Quantitative Versions of Hereditary Results on M ‐Ideals of Compact Operators

It is well–known that a Banach space X is an M –ideal in its bidual whenever the space K ( X, X ) of compact operators on X is an M –ideal in the space ℒ( X, X ) of bounded operators. The same conclusion holds whenever the space ( 1 , X ) of compact operators from 1 to X is an M –ideal in ℒ( 1 , X ). In the present paper, these results are extended by means of a unified quantitative approach describing the position of X in its bidual through the structure of ℒ( Z, Y ) when X is simultaneously isomorphic to a quotient space of Z and a subspace of Y .